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Proof step 93 from proof infinite-append-prefixnextnli (previous step)

ii(0) = 0, lf(0) = [], # lf(n0) = ii(n0), (σ from ii(n0))[n3] = nextnli0(lf(n0), n0)[n3], n3 < # nextnli0(lf(n0), n0), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextnli0(lf(m), m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2] ⊦ nextnli0(lf(n0), n0) = [], ii(n0) + # nextnli0(lf(n0), n0) = 0, nextnli0(lf(n0), n0)[n3] = σ[n3 + ii(n0)], # σ < ⌜ii(n0)⌝ + ⌜# nextnli0(lf(n0), n0)⌝ , #(σ from ii(n0)) < ⌜# nextnli0(lf(n0), n0)⌝

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: stream inst: name: fin-size-from2 seq: ⊦ ¬ # σ < ⌜n⌝ ∧ finp(σ) → #(σ from n) = ⌜(# σ).n - n⌝ subst: [n, σ] → [ii(n0), σ] flags: rewrite

Additional information

Used simplifier rules: less-fin: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; from specification nat-inf%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%stream-fin-forward: ⊦ finp(σ) → (∃ x. ⌜x⌝ = σ); (forward) from specification stream-basic%sel: ⊦ ⌜m⌝ .n = m; from specification nat-inf-basic%lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%ls-lsadd: ⊦ m < n → ⌜m⌝ < ⌜n⌝ + n∞0; from specification nat-inf%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf%ls: n0 ≤ n ⊦ m < n - n0 ↔ m + n0 < n; from specification nat%ls-02: n0 ≤ n ⊦ m ≤ n - n0 ↔ m + n0 ≤ n; from specification nat%ls-natinf-to-nat-1-2: ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; from specification nat-inf%le-ls-simp: ⊦ m ≤ n → (n < m ↔ false); from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%from-fin: ⊦ ⌜x⌝ from n = ⌜x from n⌝ ; from specification stream%sel-fin: ⊦ ⌜x⌝ [n] = x[n]; from specification stream%size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; from specification stream%inj: ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; from specification nat-inf-basic%test-02: ⊦ finp(⌜x⌝ ); from specification stream-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic1%c: ⊦ m + n = n + m; from specification nat-basic2%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%get-from-01: m + m0 < # x ⊦ (x from m)[m0] = x[m + m0]; from specification stream%lels-trans: ⊦ ¬ n∞1 < n∞ ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf%ls-natinf-to-nat-2-1: ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; from specification nat-inf%l-02: ⊦ ¬ m + n < m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%size-from-less: ⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ ); from specification stream%add-fin: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; from specification nat-inf%notfin-sizeinf: ⊦ ¬ finp(σ) → # σ = ∞; from specification stream%less-inf1: ⊦ ¬ ∞ < n∞; from specification nat-inf%stream-inf-forward: ⊦ ¬ finp(σ) → (∃ nf. ⌜nf⌝ = σ); (forward) from specification stream-basic%from-inf: ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; from specification stream%sel-inf: ⊦ ⌜nf⌝ [n] = nf(n); from specification stream%get-from-inf: ⊦ ((nf from n))(m) = nf(m + n); from specification stream Used terms: ii(n0), σ